A third strike against perfect phylogeny

TL;DR

This paper disproves a long-standing conjecture in perfect phylogeny theory, showing that for any fixed subset size, there can be incompatible sets of characters with all small subsets compatible, complicating the search for evolutionary trees.

Contribution

It proves that no fixed subset size guarantees the existence of a perfect phylogeny for all larger incompatible sets, refuting a key conjecture in the field.

Findings

Disproved the conjecture that a fixed subset size ensures perfect phylogeny.

Constructed sets of 8-state characters with no perfect phylogeny but all small subsets have one.

Highlighted implications for the complexity of phylogenetic reconstruction.

Abstract

Perfect phylogenies are fundamental in the study of evolutionary trees because they capture the situation when each evolutionary trait emerges only once in history; if such events are believed to be rare, then by Occam's Razor such parsimonious trees are preferable as a hypothesis of evolution. A classical result states that 2-state characters permit a perfect phylogeny precisely if each subset of 2 characters permits one. More recently, it was shown that for 3-state characters the same property holds but for size-3 subsets. A long-standing open problem asked whether such a constant exists for each number of states. More precisely, it has been conjectured that for any fixed integer $r$, there exists a constant $f(r)$ such that a set of $r$-state characters $C$ has a perfect phylogeny if and only if every subset of at most $f(r)$ characters has a perfect phylogeny. In this paper, we show that this conjecture is false. In particular, we show that for any constant $t$, there exists a set $C$ of $8$-state characters such that $C$ has no perfect phylogeny, but there exists a perfect phylogeny for every subset of $t$ characters. This negative result complements the two negative results ("strikes") of Bodlaender et al. We reflect on the consequences of this third strike, pointing out that while it does close off some routes for efficient algorithm development, many others remain open.